Algebraic Geometry Seminar

Let n >= 4 be an even integer and let M n be the moduli space of n points on P 1 modulo the action of PGL(2), thought of as the GIT quotient (P 1 ) n /PGL(2). The space M n has a natural projective embedding; let R n be its projective coordinate ring. The ring R n was studied classically in the context of invariant theory. In the late nineteenth centruy, Kempe proved that R n is generated by its degree one piece. Since that time however, generators for the ideal of relations have not been determined. I will talk about my recent work with Howard, Millson and Vakil towards understanding generators of this ideal. Two of our results: 1) the ideal of relations is always generated in degrees <= 3; and 2) for all n not equal to 6 the image of M n in projective space is cut out by quadrics.

Let V be a vector bundle of rank n on a variety X, with subbundles E and F of respective ranks e and f. The locus of points of X where the fibers of E and F intersect in dimension more than e+f-n is a basic example of a degeneracy locus, and it is useful to have formulas for the cohomology classes of such loci in terms of the Chern classes of E, F, and V. Many variations are possible: there should be one for each Lie type, and for each element of the corresponding Weyl group. For classical types, formulas were given by Giambelli-Thom-Porteous, Kempf-Laksov, Harris-Tu, and Fulton. In this talk, I will give formulas corresponding to exceptional type G 2 . Along the way, I'll discuss octonion bundles, and describe the G 2 flag variety in concrete, linear-algebraic terms.

Let G be a semisimple Lie group and which is acting on X. Then we consider X//G and X//T where T is a maximal torus of G . We will explain a comparison theory of orbifold cohomologies of X//G and X//T when both spaces are orbifolds.